Solving problems involving forces

  1. Free-body diagrams are a must when solving problems involving forces.
    1. Start with fundamental forces that will always exist; i.e., gravitational and electromagnetic forces.
    2. Add forces that are given or implied by the problem itself.
    3. Depending on the context of the problem, add any meaningful reactionary forces.
    4. If you have more than one FBD for objects interacting with one another, always consider Newton's third law.
    5. The only non-force quantity that may be indicated on the FBD is acceleration. Do not attach it to the body, but rather draw it as a visually distinct arrow (typically a squiggly-line arrow that is not attached to the object).
    6. Have a good look at your FBD before continuing further. See if it makes sense given the acceleration you may know from the context of the problem.
  2. Dimensions. If your problem is in more than a single dimension, you need to choose two (or three if it is a three-dimensional problem) perpendicular dimensions to decompose your vectors in. If you know the direction of acceleration, it is most convenient to use that as one of your dimensions. This makes acceleration zero in all other dimensions.
  3. Sign convention. In each dimension you have, you should clearly indicate which direction is chosen as positive. Forces are vectors, and direction is an fundamental property of a vector. It is easiest and safest to take the direction of acceleration (if it exists) as a positive direction.
  4. Vectors are analyzed in each dimension independently from the other dimensions. Remember that scalars (e.g., time) are the same in all dimensions, and are usually used as links between those dimensions.
  5. Force analysis in each dimension
    1. Start with determining whether Newton's first law (\(F_{net}=0\)) or second law (\(F_{net}=ma\)) applies to the situation in this dimension.
    2. Replace \(F_{net}\) with the sum of the individual forces in this dimension. Be sure to incorporate the correct sign at this step.
    3. If you can do any symbolic substitutions, do it at this point. Some of the most famous substitutions are \(F_g=mg\) and \(F_f=\mu F_N\). Think before you do this step, because not every possible symbolic substitution is necessarily a useful one in every situation.
    4. At this point, you may have a system of equations. Identify in them
      • the required;
      • the givens or knowns (including those values that can be calculated from the givens); and
      • quantities that are still unknown, but are not required. We want to eliminate those quantities from our equations.
      Consider solving the system of equation by substitution or elimination.
    5. Finally, substitute into the equation with the givens/knowns to calculate the required.
  6. You may need to combine your findings from different dimensions to find the final solution. For example, you may need to calculate tension that has two components in different dimensions. You will need to add the components as vectors and determine the direction of the resultant.
  7. A final statement must be provided, and it must reflect the interpretation of your numbers. No solution is complete without proper interpretation.